Efficiency of Feynman's quantum computer

Feynman's circuit-to-Hamiltonian construction enables the mapping of a quantum circuit to a time-independent Hamiltonian. This model introduces a Hilbert space made from an ancillary clock register tracking the progress of the computation. In this paper, we explore the efficiency, or run-time, of a quantum computer that directly implements the clock system. This relates to the model's probability of computation completion which we investigate at an established optimal time for an arbitrary number of gates $k$. The relationship between the run-time of the model and the number of gates is obtained both numerically and analytically to be $O(k^{5/3})$. In principle, this is significantly more efficient than the well investigated Feynman-Kitaev model of adiabatic quantum computation with a run-time of $O(k^4)$. We address the challenge which stems from the small window that exists to capture the optimal stopping time, after which there are rapid oscillations of decreasing probability amplitude. We establish a relationship for the time difference between the first and second maximum which scales as O($k^{1/3}$).